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### Problems: Discrete Probability Distributions Part 1

1. John has a special die that has one side with a six, two sides with twos and three sides with ones. He offers you the following game. He will throw the die and will pay you in dollars the number that comes up. (If it comes up six, you get \$6.00, if it come up one, you get \$1, etc.) He will charge you \$2.00 to play this game. Is this a good game to play for you? Explain by computing the expected value of this game.

 x p(x) 1 3/6 2 2/6 6 ?

2.

 Outcome: Probability: 0 ? 1 .1 2 .2 3 .5

a) What value should replace the question mark if this is a probability distribution?
b) Is this a discrete or continuous distribution? Explain.
c) What is the mean of this probability distribution?
d) What is the standard deviation of this probability distribution?

a) The two sequences are equally likely.
b) The first sequence is twice as likely as the second.
c) The first sequence is four times as likely as the second.
d) The first sequence is eight times as likely as the second.
e) The first sequence is sixteen times as likely as the second.

4. A game of chance is played by spinning a wheel and paying the amount that comes up. There are four possible outcomes:

 x p(x) \$1 .50 \$2 .30 \$5 .15 \$10 ??
a) What is the probability that \$10 comes up?
b) What is the expected value of this game?

5a) Show that P(x) = x/11 - .25 for x = 4, 5, 6, 7 is a probability distribution.

b) Find the expected value of this distribution.
c) Find the variance of this distribution.
d) Prove that P(x) = x/11 - .25 for x = 3, 4, 5, 6, is NOT a probability distribution.

6. Jack has a trick coin that will come up heads two-thirds of the time.

a) What is the probability that it will come up tails?
b) John offers you the following game. If the coin comes up tails, he pays you \$1.50. If it comes up heads, you pay him \$1.00. How much on the average will you win or lose each time you play this game (or what is the expected value of this game)?

7. Here is an example of a discrete, uniform probability distribution:

 x p(x) 0 .2 1 .2 2 .2 3 .2 4 .2

Find:

(0 < P(x) < 4) = ________
(P(x) > 1) = ________
(P(x) = 1) = ________
(P(x) < 1) = _________

What is the expected value of x?
What is the standard deviation of this distribution?

8. The game show Deal or No Deal features 26 suitcases with various amounts of money. The contestant chooses one than then begins to open the others. At the end of each round, the "Banker" makes an offer to end the game. The game ends either when the player takes the offer, or when he or she has opened all 25 suitcases that he or she did not choose, in which case he or she gets the amount in the chosen suitcase.

Suppose that the player has four suitcases left with the following amounts in them: \$1,000,000, \$100,000, \$1000, and \$100. What would the fair offer for the contestant to quit be from the "Banker"?

9. P(x) = .5 -x/10 is a probability distribution for x = 1, 2, 3, 4.

a) Suppose we use this rule to provide two outcomes. If the outcomes are independent, what is the probability that the first one will be a 4 and the second will be a 1?
b) What is the probability that at least one of the two outcomes will be a 1?